We present an importance sampling algorithm that can produce realisations of Markovian epidemic models that exactly match observations, taken to be the number of a single event type over a period of time. The importance sampling can be used to construct an efficient particle filter that targets the states of a system and hence estimate the likelihood to perform Bayesian inference. When used in a particle marginal Metropolis Hastings scheme, the importance sampling provides a large speed-up in terms of the effective sample size per unit of computational time, compared to simple bootstrap sampling. The algorithm is general, with minimal restrictions, and we show how it can be applied to any continuous-time Markov chain where we wish to exactly match the number of a single event type over a period of time.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Also known as the Gillespie algorithm or the Doob-Gillespie algorithm.
In fact this can be done analytically for this model.
This condition may not be true after the final observed infection event, depending on what other observations are made on the system afterwards.
Aho, A.V., Ullman, J.D.: Foundations of Computer Science. W. H. Freeman and Company, New York (1995)
Anderson, D.F.: A modified next reaction method for simulating chemical systems with time dependent propensities and delays. J. Chem. Phys. 127, 214107 (2007)
Andrieu, C., Doucet, A., Holenstein, R.: Particle Markov chain Monte Carlo methods (with discussion). J. R. Stat. Soc. B 72, 269–342 (2010)
Black, A.J., Geard, N., McCaw, J.M., McVernon, J., Ross, J.V.: Characterising pandemic severity and transmissibility from data collected during first few hundred studies. Epidemics 19, 61–73 (2017)
Black, A.J., McKane, A.J.: Stochastic formulation of ecological models and their applications. Trends Ecol. Evol. 27, 337–345 (2012)
Black, A.J., Ross, J.V.: Estimating a Markovian epidemic model using household serial interval data from the early phase of an epidemic. PLoS ONE 8, e73420 (2013)
Black, A.J., Ross, J.V.: Computation of epidemic final size distributions. J. Theor. Biol. 367, 159–165 (2015)
David, H.A., Nagaraja, H.N.: Order Statistics, 3rd edn. Wiley, New York (2005)
Del Moral, P., Jasra, P., Lee, A., Yau, C., Zhang, X.: The alive particle filter and its use in particle Markov chain Monte Carlo. Stoch. Anal. Appl. 33, 943–974 (2015)
Doucet, A., de Freitas, N., Gordon, N.J. (eds.): Sequential Monte Carlo Methods in Practice. Springer, New York (2001)
Doucet, A., Johansen, A.M.: A tutorial on particle filtering and smoothing: fifteen years later. Handb. Nonlinear Filter. 12(656–704), 3 (2009)
Doucet, A., Pitt, M.K., Deligiannidis, G., Kohn, R.: Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator. Biometrika 102, 295–313 (2015). https://doi.org/10.1093/biomet/asu075
Drovandi, C.C.: Pseudo-marginal algorithms with multiple CPUs (2014). http://eprints.qut.edu.au/61505
Drovandi, C.C., McCutchan, R.A.: Alive SMC\(^2\): Bayesian model selection for low-count time series models with intractable likelihoods. Biometrics 72, 344–353 (2016)
EpiStruct.: (2017). https://github.com/EpiStruct/Black-2018
Gibson, G.J., Renshaw, E.: Estimating parameters in stochastic compartmental models using Markov chain methods. Math. Med. Biol. 15, 19–40 (1998). https://doi.org/10.1093/imammb/15.1.19
Gibson, M.A., Bruck, J.: Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. A 104, 1876–1889 (2000)
Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434 (1976)
Golightly, A., Kypraios, T.: Efficient SMC\(^2\) schemes for stochastic kinetic models. Stat. Comput. (2017). https://doi.org/10.1007/s11222-017-9789-8
Golightly, A., Wilkinson, D.J.: Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo. Interface Focus 1, 807–820 (2011). https://doi.org/10.1098/rsfs.2011.0047
Gordon, N.J., Salmond, D.J., Smith, A.F.M.: Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F 140, 107–113 (1993)
Jenkinson, G., Goutsias, J.: Numerical integration of the master equation in some models of stochastic epidemiology. PLoS ONE 7, e36160 (2012)
Jewell, C.P., Kypraios, T., Neal, P., Roberts, G.O.: Bayesian analysis for emerging infectious diseases. Bayesian Anal. 4, 465–496 (2009). https://doi.org/10.1214/09-BA417
Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton, NJ (2007)
Knuth, D.: The Art of Computer Programming, vol. 1. Addison-Wesley, Reading, MA (1997)
Kroese, D.P., Taimre, T., Botev, Z.I.: Handbook of Monte Carlo Methods. Wiley, New York (2011)
Lau, M.S.Y., Cowling, B.J., Cook, A.R., Riley, S.: Inferring influenza dynamics and control in households. Proc. Natl. Acad. Sci. 112, 9094–9099 (2015)
McKinley, T.J., Ross, J.V., Deardon, R., Cook, A.R.: Simulation-based Bayesian inference for epidemic models. Comput. Stat. Data Anal. 71, 434–447 (2014). https://doi.org/10.1016/j.csda.2012.12.012
O’Neill, P.D., Roberts, G.O.: Bayesian inference for partially observed stochastic epidemics. J. R. Stat. Soc. A 162, 121–130 (1999). https://doi.org/10.1016/j.epidem.2013.12.002
Pitt, M.K., Silva, R., Giordani, P., Kohn, R.: On some properties of Markov chain Monte Carlo simulation methods based on the particle filter. J. Econom. 171, 134–151 (2012). https://doi.org/10.1016/j.jeconom.2012.06.004
Pooley, C.M., Bishop, S.C., Marion, G.: Using model-based proposals for fast parameter inference on discrete state space, continuous-time Markov processes. J. R. Soc. Interface 12, 20150225 (2015)
Regan, D.G., Wood, J.G., Benevent, C., et al.: Estimating the critical immunity threshold for preventing hepatitis a outbreaks in men who have sex with men. Epidemiol. Infect. 144, 1528–1537 (2016). https://doi.org/10.1017/S0950268815002605
Roh, M.K., Gillespie, D.T., Petzold, L.R.: State-dependent biasing method for importance sampling in the weighted stochastic simulation algorithm. J. Chem. Phys. 133, 174106 (2010)
Sherlock, C., Thiery, A.H., Roberts, G.O., Rosenthal, J.S.: On the efficiency of pseudo-marginal random walk metropolis algorithms. Ann. Stat. 43, 238–275 (2015)
Stockdale, J.E., Kypraios, T., O’Neill, P.D.: Modelling and bayesian analysis of the Abakaliki smallpox data. Epidemics 19, 13–23 (2017). https://doi.org/10.1016/j.epidem.2016.11.005
van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. Elsevier, Amsterdam (1992)
Walker, J.N., Ross, J.V., Black, A.J.: Inference of epidemiological parameters from household stratified data. PLoS ONE 12, e0185910 (2017)
This research was supported by an ARC DECRA fellowship (DE160100690). AJB also acknowledges support from both the ARC Centre of Excellence for Mathematical and Statistical Frontiers (CoE ACEMS), and the Australian Government NHMRC Centre for Research Excellence in Policy Relevant Infectious diseases Simulation and Mathematical Modelling (CRE PRISM\(^2\)). Supercomputing resources were provided by the Phoenix HPC service at the University of Adelaide. AJB would also like to thank Joshua Ross and James Walker for comments on an earlier draft of the manuscript.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
About this article
Cite this article
Black, A.J. Importance sampling for partially observed temporal epidemic models. Stat Comput 29, 617–630 (2019). https://doi.org/10.1007/s11222-018-9827-1
- Importance sampling
- Markov chain
- Epidemic modelling
- Particle filter